PSI - Issue 74

Dragan Pustaić et al. / Procedia Structural Integrity 74 (2025) 70 – 76 Dragan Pustaić / St ructural Integrity Procedia 00 (20 2 5) 000 – 000

71

2

1. Introduction The straight crack of the length, 2 a , according to the Fig. 1.a, is incorporated in the thin, infinite plate. The crack is loaded by uniformly distributed, continuous loading, p 0 , which acts in-plane of a plate and opens the crack. The load is monotonously increased. All other plate edges are free of loads. It is assumed that the plate is made of ductile metallic material which is homogeneous and isotropic. At plastic deformation, the plate material is non-linearly hardened, which is good described by the Ramberg-Osgood´s analytical expression. The small plastic zones of the length, r p , according to Figs. 1.a and 1.b, are formed around the crack tips, when loading the plate. These zones are increased with increasing of the loads, p 0 . According to the Dugdale´s idea, these small plasticized areas can be added to the real, physical crack and together make the fictitious elastic crack of the length 2 b = 2∙( a + r p ), Figs. 1.a and 1.b. The non-linearly distributed cohesive stresses , p ( x ), act on a part, r p , of a fictitious elastic crack, according to the Fig. 1b. The normal stress, σ y , will not be singular within the tip of the fictitious elastic crack, but it will have an ultimate value. The non-singularity stress condition at the point x = b , means that the resulting stress intensity factor will be equal to zero, i.e. K = K ext + K coh = 0. That condition was used in the elastic plastic crack analysis in numerous papers, for example Chen et al. (1992), Guo (1993), Neimi tz (2000), Pustaić et al. (2019), and so on. Nomenclature

longitudinal strain, -

ε

half physical crack length, m half fictitious crack length, m

a

strain corresponds to the yield stress according to Hooke´s law, - strain hardening exponent, - uniformly distributed, continuous load acts over the crack surface, MPa material parameter, -

0 ε

b h

thickness of a plate, m

α

plastic zone magnitude around crack tip Young´s modulus of elasticity, GPa the two pairs of the concentrated forces acting on the crack surface, N/m non-dimensional loading of the crack, - stress intensity factor, (SIF), from the external loading, MPa m stress intensity factor from the cohesive stresses, MPa m

r p E

n p 0

cohesive stress, MPa independent variable, m

F

p (x)

F /( σ 0 ∙ a )

x

m (x, b)

Green´s function,

1 2 m −

K ext

( ) x Γ

Gamma function, -

K coh

yield stress, MPa

(

)

0 σ

; ; ; z α β γ

2 1 F

Hypergeometric function, -

independent variable in which the plastic zone is incorporated, -

(

normal stress, MPa

p =   ⋅ + 2

σ y

P r

a r

p

The cohesive stresses, p ( x ), are distributed according to the analytical expression which distribution is unknown in advance. The process of deriving that expression is described in the pape r Pustaić et al. (2006) . The expression suggested by the authors Hoffman and Seeger (1985) will be used in this article. It has shown as a very good and accurate approximation of the real stress distribution and equals

) ( ) 1 1 n + 

( )

(

0 p r x a σ 

.

(1)

p x

= ⋅ 

− 

The cohesive stress, p ( x ), [MPa], according to that expression, is a function of two parameters, the plastic zone magnitude, r p , [m], and the strain-hardening exponent, n .